In geological instrumentation, it is frequent to find the "integration of the acceleration" measurement, as the "empirical estimate" of the velocity, namely, the Particle Point Velocity (PPV).
Assuming:
- An real physical acceleration and velocity magnitudes -$a_0(t)$ and v_0(t)- for a steadily oscillating system, unknown. This is standard on sensors placed on ground. This is not the case of aeronautics and navigation necessarily,
- An electrical additive noise e(t)~N(0,\sigma), unknown,
- An acceleration measurement, a(t), known,
- A sampling time \tau over time, functions and integrals (just for treating the process discretely, not required though).
An accelerometer register can be simply modeled as:
$$a(t)=a_0(t)+e(t)$$
Thus the velocity estimate would be:
$$v(t)=\int_0^ta_0(\tau)d\tau+\int_0^te(\tau)d\tau=v_0(t)+\int_0^te(\tau)d\tau$$
For cancelling the "noisy" tendency in v(t), a "baseline correction" procedure is often stated as the "correct" solution. Which is nothing more than fitting a customary function to the wiener process.
This is a "Dead Reckoning" process.
**Which should be the optimal way for estimating v_0(t)? from a mathematical point of view**.
PS: The estimate at the instant t, of a Wiener Process with conditional deviation \sigma is:
\$w(t)\$~\$N(0,\sigma \sqrt t)\$
Hence, if the estimated resolution -from DAQ resolution- of the velocity measurement is \nu, after the time (\frac{\nu}{\sigma})^2 the integrated noise will surpass the sensor resolution with "1-sigma" confidence, invalidating the procedure.
Actually, after the time \$t\$ the wiener process itself will be anywhere in 1-sigma confidence interval (-\sigma\sqrt t,\sigma\sqrt t).
IS there an standard method, taking in place the inherent gaussian integrated error into account, available?.
I am trying to solve -still just trying- this problem by imposing:
- The velocity has no real "baseline" - i.e. a "real" velocity measurement will behave like an accelerometer measure, centered at zero.
- The velocity at every smaller interval should have no "baseline". This is incorrect at very small intervals, but a potential solution.
- An easy solution is to take any smoother filter, with a look&feel filtering power. This way a trend is estimated, removing the random walk noise and leaving a gaussian noise. This is the solution i have at this moment.
- Standard solutions like butterworth lowpass filters as baseline correction, though proper, are not correct, because the butterworth are meant for removing specific low frequencies, and the Wiener process is a ramp in PSD.
- A potential solution could be mimic the Wiener PSD, and apply that filtering.
Thanks in advance,
Assuming:
- An real physical acceleration and velocity magnitudes -$a_0(t)$ and v_0(t)- for a steadily oscillating system, unknown. This is standard on sensors placed on ground. This is not the case of aeronautics and navigation necessarily,
- An electrical additive noise e(t)~N(0,\sigma), unknown,
- An acceleration measurement, a(t), known,
- A sampling time \tau over time, functions and integrals (just for treating the process discretely, not required though).
An accelerometer register can be simply modeled as:
$$a(t)=a_0(t)+e(t)$$
Thus the velocity estimate would be:
$$v(t)=\int_0^ta_0(\tau)d\tau+\int_0^te(\tau)d\tau=v_0(t)+\int_0^te(\tau)d\tau$$
For cancelling the "noisy" tendency in v(t), a "baseline correction" procedure is often stated as the "correct" solution. Which is nothing more than fitting a customary function to the wiener process.
This is a "Dead Reckoning" process.
**Which should be the optimal way for estimating v_0(t)? from a mathematical point of view**.
PS: The estimate at the instant t, of a Wiener Process with conditional deviation \sigma is:
\$w(t)\$~\$N(0,\sigma \sqrt t)\$
Hence, if the estimated resolution -from DAQ resolution- of the velocity measurement is \nu, after the time (\frac{\nu}{\sigma})^2 the integrated noise will surpass the sensor resolution with "1-sigma" confidence, invalidating the procedure.
Actually, after the time \$t\$ the wiener process itself will be anywhere in 1-sigma confidence interval (-\sigma\sqrt t,\sigma\sqrt t).
IS there an standard method, taking in place the inherent gaussian integrated error into account, available?.
I am trying to solve -still just trying- this problem by imposing:
- The velocity has no real "baseline" - i.e. a "real" velocity measurement will behave like an accelerometer measure, centered at zero.
- The velocity at every smaller interval should have no "baseline". This is incorrect at very small intervals, but a potential solution.
- An easy solution is to take any smoother filter, with a look&feel filtering power. This way a trend is estimated, removing the random walk noise and leaving a gaussian noise. This is the solution i have at this moment.
- Standard solutions like butterworth lowpass filters as baseline correction, though proper, are not correct, because the butterworth are meant for removing specific low frequencies, and the Wiener process is a ramp in PSD.
- A potential solution could be mimic the Wiener PSD, and apply that filtering.
Thanks in advance,
